Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring II, Quasilocal Case

The rings we consider in this article are commutative with identity $1\neq 0$ and are not fields. Let $R$ be a ring. We denote the collection of all proper ideals of $R$ by $\mathbb{I}(R)$ and the collection $\mathbb{I}(R)\setminus \{(0)\}$ by $\mathbb{I}(R)^{*}$. Let $H(R)$ be the graph associated with $R$ whose vertex set is $\mathbb{I}(R)^{*}$ and distinct vertices $I, J$ are adjacent if and only if $IJ\neq (0)$. The aim of this article is to discuss the planarity of $H(R)$ in the case when $R$ is quasilocal.